The application of finding eigenstates for solving differential equations

Hello,
I'm trying to understand a little deeper the method I'm using to solve my diffusion problem. Does method of eigenstates for solving differential equation only work for continuous functions and if so what is an explanation for this?

What "continuous functions" are you talking about? The solutions? (If a function satisfies a differential equation it must be differentiable and so must be continuous) Or the coefficients? Non-continuous functions can lead to very complicated things like "shocks" which cannot typically be written in terms of "eigenstates"- though you can solve the problem on different "sides" of the shock lines.

Thanks for your response. I'm looking at a diffusion problem in a porous solid for which I've been advised to change the subect of the diffusion equation from concentration of the diffusing species to the chemical potential which is a continuous function over all of the system space (whereas the concentration meets a boundary at the pore-edge) - and by doing this it becomes possible to use the method of eigenstates to calculate the chemical potential.
I guess this fits with what you said about shocks - is this essentially what occurs for the concentration at the pore boundary?